Solve : `1+2cosec x=-("sec"^(2)(x)/(2))/(2)`

Answer

`x=2n pi-(pi)/(2), n in Z`

Explanation

`1+2 cosec x =-("sec"^(2)(x)/(2))/(2)`
`rArr 1+2((1+"tan"^(2)(x)/(2))/(2tan.(x)/(2)))=-((1+"tan"^(2)(x)/(2)))/(2)`
`rArr 2tan.(x)/(2)+2+2"tan"^(2)(x)/(2)+tan.(x)/(2)+"tan"^(3)(x)/(2)=0`
`rArr "tan"^(3)(x)/(2)+2"tan"^(2)(x)/(2)+3tan.(x)/(2)+2=0`
`rArr (tan.(x)/(2)+1)("tan"^(2)(x)/(2)+tan.(x)/(2)+2)=0`
`tan.((x)/(2))+1=0`
`rArr tan.(x)/(2)=tan(-(pi)/(4))`
`rArr x = 2n pi-(pi)/(2), n in Z`